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12

The Singular Value Decomposition and ItYs Applications in Image Processing

Feb 11, 2022

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Page 1: The Singular Value Decomposition and ItYs Applications in Image Processing

Aüi 5ð?}ý*@h V@*ýi #iUL4TLtð|ðL? @?_ W|<t

úTT*ðU@|ðL?t ð? W4@}i èhLUittð?}

çühðt|LTüih a@tL? ï}_i?c A@44ði Oýg�@|ü eD Q çL**i}i Lu |üi +i_êLL_t

#iUi4Mih �Hc �bb.

úMt|h@U|

i ýti tð?}ý*@h ñ@*ýi _iUL4TLtð|ðL? |L @TThL ð4@|i *@h}ic ý?4@?@}ií@M*i 4@|hðUit ð?|L t4@**ih ð?ñih|ðM*i t^ý@hi 4@|hðUitë ALTðUt ð?U*ý_i |üi4@|üi4@|ðUt Miüð?_ |üi tð?}ý*@h ñ@*ýi _iUL4TLtð|ðL?c |üi ýti Lu }>z'>Kêð|ü |üi tð?}ý*@h ñ@*ýi _iUL4TLtð|ðL?c @?_ |üi @TT*ðU@|ðL? Lu |üi tð?}ý*@hñ@*ýi _iUL4TLtð|ðL? |L ð4@}i UL4ThittðL? êð|üLý| tð}?ð�U@?| _@|@ *Lttë

� W?|hL_ýU|ðL?

Wkh vlqjxodu ydoxh ghfrpsrvlwlrq +VYG,/ rqh ri wkh prvw xvhixo wrrov ri olqhdudojheud/ lv d idfwrul}dwlrq dqg dssur{lpdwlrq whfkqltxh zklfk hòhfwlyho| uhgxfhvdq| pdwul{ lqwr d vpdoohu lqyhuwleoh dqg vtxduh pdwul{1 Wkh VYG surylghvzkhuh rwkhu olqhdu dssur{lpdwlrq whfkqltxhv idlo1 Zlwk doprvw hyhu| wrro dqgwhfkqltxh glvfxvvhg lq olqhdu dojheud/ wkhuh lv d gholplwhu/ _111surylghg wkdw wkhpdwul{ lv erwk lqyhuwleoh dqg vtxduh1% Wkh vlqjxodu ydoxh ghfrpsrvlwlrq qrwrqo| dssur{lpdwhv wklv vshfldo fdvh vfhqdulr/ exw lw zloo dovr zrun lw*v pdjlf rqhyhu| rwkhu srvvleoh vfhqdulr1 Wkh VYG zrunv zrqghuixoo| zlwk erwk xqghu0dqg ryhu0 ghwhuplqhg pdwulfhv1

2 �@|üi4@|ðUt îiüð?_ |üi 5V#

Wkhruhp 4 Wkhuh h{lvwv pdwulfhv X / G/ Y vxfk wkdw wkh pdwul{ D/ dq p û qpdwul{ zlwk udqn u/ fdq eh idfwruhg lqwr wkh irup D @ XGY W > zkhuh G lv d_gldjrqdo% p û q pdwul{ zlwk uhdo hqwulhv

â4 è â5 è ü ü ü è âu è 3/ u @ plq+p>q,dqg X )Y duh ruwkrjrqdo pdwulfhv vxfk wkdw X lv dq p û p pdwul{ dqg Y lv dqq û q pdwul{1 Wkh gldjrqdo hqwulhv ri G/ iâ4> â5> ü ü üâuj duh fdoohg wkh vlqjxoduydoxhv ri wkh pdwul{ D/ wkh froxpqv ri wkh pdwul{ X duh fdoohg wkh ohiw vlqjxoduyhfwruv ri wkh pdwul{ D/ dqg wkh froxpq vsdfh ri wkh pdwul{ Y +ru wkh urz vsdfh

4

Page 2: The Singular Value Decomposition and ItYs Applications in Image Processing

ri Y W , duh fdoohg wkh uljkw vlqjxodu yhfwruv ri wkh pdwul{ D1 Ixuwkhupruh/ wkhpdwul{ D fdq eh frqghqvhg dv Du @ Xu÷Yu/ zkhuh ÷ lv dq u û u gldjrqdo pdwul{zlwk hqwulhv iâ4>â5>===> âuj1 Ixuwkhu vwloo/ wkh pdwul{ D fdq eh dssur{lpdwhg dvDl @ Xl÷lYl

W / zkhuh l fruuhvsrqgv wr wkh ñuvw l urzv dqg l froxpqv ri hdfklqglylgxdo pdwul{ D/ X / ÷/ Y 1

Surri1 Jlyhq wkdw D lv dq p û q pdwul{ lq wkh irup

D @k $d4

$d5 ü ü ü $

dl ü ü ü $dq

lvxfk wkdw

$dl @

59997dl4dl5111dlp

6:::8Wkhuhiruh

D @

59997d44 d45 ü ü ü d4l ü ü ü d4qd54 d55 ü ü ü d5l ü ü ü d5q111

111111

111dp4 dp5 ü ü ü dpl ü ü ü dpq

6:::82ë� wð?i@h Ah@?tuLh4@|ðL?t

Wkh surfhvv ri pxowlso|lqj dq| w|slfdo yhfwru${ e| d pdwul{ D lv nqrzq dv

suhiruplqj d olqhdu wudqvirupdwlrq ri${ 1 Li zh xvh wkh qrwdwlrq W +

${, @ D

${ /

zkhuh W +${, pdsv dq| yhfwru

${ rqwr wkh yhfwru D

${ dqg lv ghqrwhg e|

${ $ D

${ / doo wkdw lv uhtxluhg wr pd{lpl}h wkh wudqvirupdwlrq W +

${,/ lv wr pd{lpl}h wkh

txdqwlw| D${ 1

H{huflvh 5 Li è lv dq hljhqydoxh ri wkh pdwul{ D/ dqg${ lv wkh fruuhvsrqglqj

xqlw hljhqyhfwru/ ñqg wkh xqlw hljhqyhfwru dw zklfk mmD${ mm lv pd{lpl}hg1

Vroxwlrq Vlqfh mmD${ mm5 lv pd{lpl}hg iru wkh vdph ${ dv lv mmD${ mm/ dqg vlqfhmmD${ mm5 lv hdvlhu wr h{dplqh/ wklv vxevwlwxwlrq lq wkh sureohp zloo eh pdgh1

mmD${ mm5 @ +D${,W +D

${,

@${WDWD

${

@${W+DWD,

${ +4,

+DWD, lv d v|pphwulf pdwul{ dv lv vkrzq1

+DWD,W @ DWDWW @ DWD

5

Page 3: The Singular Value Decomposition and ItYs Applications in Image Processing

Wkhuhiruh wkh sureohp kdv ehfrph wr pd{lpl}h wkh txdgudwlf irup${WV${ /

zkhuh V lv dq lqyhuwleoh vtxduh pdwul{ dqg${ lv dv ehiruh1 Lw lv nqrzq wkdw wkh

xqlw hljhqyhfwru$y4 fruuhvsrqglqj wr wkh vwulfwo| grplqdqw hljhqydoxh â4 ri wkh

pdwul{ V pd{lpl}hv wklv htxdwlrq1 Wklv phdqv wkdw mmD${ mm lv pd{lpl}hg iru${ @

$y41

D uhodwlrqvkls ehwzhhq â dqg è fdq eh rewdlqhg e| wdnlqj wklv vwhs mxvwvoljkwo| iduwkhu1 Zh qrwlfh wkdw

D${ @ è

${

+DWD,$y @ â

$y

Wkhuhiruh

mmD${ mm @ mmè${ mm@ mèm mm${ mm@ mèm

Dqg vxevwlwxwlqj edfn lqwr htxdwlrq +4,/

mèm5 @${W+DWD,

${

${ mèm5 @

${${W+DWD,

${

mèm5${ @ +DWD,${

exw${ @

$y / vr

mèm5$y @ â$y

mèm5$y$y W @ â$y$yW

mèm5 @ â

2ë2 W?ñih|ðM*i 5^ý@hi �@|hð

Ohw V @ +DWD, vxfk wkdw V lv dq q û q lqyhuwleoh vtxduh pdwul{ lq wkh irup

V @k $v4

$v5 ü ü ü $

vl ü ü ü $vq

lzkhuh

$vl @

59997vl4vl5111vlq

6:::8

6

Page 4: The Singular Value Decomposition and ItYs Applications in Image Processing

Wkhuhiruh V lv dq q û q pdwul{ lq wkh irup

V @

59997v44 v45 ü ü ü v4qv54 v55 ü ü ü v5q111

111111

vq4 vq5 ü ü ü vqq

6:::8Zh kdyh mxvw vkrzq wkdw mmD${ mm lv pd{lpl}hg iru ${ @ $

y4 zkhuh$y4 lv wkh

hljhqyhfwru ri V fruuhvsrqglqj wr wkh vwulfwo| grplqdqw hljhqydoxh â41 Vlqfhi$y4>$y5> ===> $yqj lv dq ruwkrqrupdo vhw ri wkh xqlw hljhqyhfwruv ri V/ wkhq wkhpdwul{ Y @ ^

$y4>

$y5> ===>

$yq` lv dq q û q ruwkrjrqdo pdwul{ frpsulvhg ri wkh xqlw

hljhqyhfwruv ri V lq rughu ri ghvfhqglqj grplqdqfh1 Lq dgglwlrq/ wkh hljhqydoxhviâ4> â5> ===> âqj ri V duh sxw lqwr wkh cgldjrqdo* hqwulhv ri wkh sdgghg pdwul{ G1Lw lv dw wklv srlqw wkdw zh pxvw uhphpehu wkh jrdo zklfk zh duh wu|lqj wr

dfklhyh1 Wklv lv/ wkdw jlyhq dq| p û q pdwul{ D/ wkhuh h{lvwv d idfwrul}dwlrqD @ XGY W 1 Zh nqrz wkh irupdw ri G/ Y / dqg D/ vr d pdwul{ X fdq ehrewdlqhg lq whupv ri wkh uhpdlqlqj wkuhh pdwulfhv lq wkh yhu| ohdvw1

D @ XGY W +5,

X @ DYGý4

Zh fdq frqvlghu wkh pdwul{ G dv dq p û q pdwul{ lq wkh irup

G @k $g4

$g5 ü ü ü

$gl ü ü ü

$gq

lzkhuh wkh yhfwru

$gl lv vxfk wkdw wkh rqo| qrq0}hur hqwu| ri wkh yhfwru lv wkh l

whup/ zklfk lv wkh lqyhuvh ri wkh l |ü vlqjxodu ydoxh ri wkh pdwul{ D1 Qrz wkhsurgxfw Y Gý4 fdq eh h{suhvvhg dv

Y Gý4 @k $y4

$y5 ü ü ü $

yq

l59999997

$g4ý4

$g5ý4

111$gqý4

6::::::8@

k4â4

$y4

4â5

$y5 ü ü ü 4

âq

$yq

lWkhuhiruh Y Gý4 lv dq q û q1 Wkh surgxfw ri wkh ohiw pxowlsolfdwlrq ri D/ dqp û q pdwul{/ wr Y Gý4/ dq q û p pdwul{/ |lhogv wkdw X lv dq p û p pdwul{lq wkh irup

X @

é4

â4D$y4>

4

â5D$y5> ===>

4

âqD$yq

è

7

Page 5: The Singular Value Decomposition and ItYs Applications in Image Processing

2ëô çL?tL*ð_@|ð?} @ �@|hð Ntð?} 5V#

Ghñqlwlrq 6 Wkh udqn ri d pdwul{ lv wkh qxpehu ri olqhduo| lqghshqghqw yhfwruvlq wkh froxpq vsdfh ri wkh pdwul{1

H{huflvh 7 Ohwwlqj D/ X / G/ Y eh dv ehiruh/ vkrz wkdw li wkh udqn ri wkh pdwul{lv ohvv wkdq wkh vpdoohvw glphqvlrq ri wkh pdwul{/ d vpdoohu pdwul{ Du fdq ehfrpsxwhg1

Vroxwlrq Li zh ohw u @ udqn+D, dqg ghñqh ÷ wr eh dq u û u gldjrqdopdwul{ zkhuh wkh pdlq gldjrqdo ri wkh pdwul{ duh wkh ñuvw u hljhqydoxhv ri wkhpdwul{ V lq rughu ri ghvfhqglqj grplqdqfh/ wkhq ÷ zrxog eh lq wkh irup

÷ @

59997â4 3

â51 1 1

3 âu

6:::8dqg wkh ydoxhv iâ4> â5> ===> âuj zrxog eh fdoohg wkh vlqjxodu ydoxhv ri wkh pdwul{D1 Li zh ghñqh wkh pdwul{ G wr eh dq p û q pdwul{ zlwk frqwdlqlqj ÷ dqgdxjphqwhg zlwk }hurv vr dv wr jlyh lw wkh vdph glphqvlrqv dv wkh pdwul{ D/ Gzrxog eh lq wkh irup

G @

é÷ 33 3

è+qý u, froxpqv ri }hurv

+pý u, urzv ri }hurv

Li zh ohw Yu @ ^$y4>

$y5> ===>

$yu` dqg ohw wkh uhpdlqlqj yhfwruv

q $yu.4>

$yu.5> ===>

$yqr

eh wkh edvlv iru wkh pdwul{ Yqýu/ wkhq Y fdq eh zulwwhq dv

Y @ ^Yu> Yqýu`

D vlplodu dujxphqw fdq eh pdgh iru X 1

X @ ^Xu> Xpýu`

Wkhuhiruh/ lw fdq eh vkrzq wkdw wkh pdwul{ D fdq eh zulwwhq dv d vpdoohu u ûu pdwul{ Du1

D @ XGY W

D @ ^Xu> Xpýu`é÷ 33 3

è éYu

W

Yqýu W

èD @ Xu÷Yu

W @ Du

8

Page 6: The Singular Value Decomposition and ItYs Applications in Image Processing

2ëe úTThL ð4@|ð?} @ �@|hð Ntð?} 5V#

Lw fdq eh vkrzq iurp wkh suhylrxv h{dpsoh wkdw wkh pdwul{ D zklfk htxdov wkhpdwul{ Du fdq eh h{suhvvhg dv d olqhdu frpelqdwlrq ri wkh vlqjxodu ydoxhv ri wkh

pdwul{ D dqg wkh fruuhvsrqglqj yhfwruv$xl dqg

$ylW

D @ XGY W

D @$x4

$g4$y4W.

$x5

$g5$y5W. ü ü ü. $

xl$gl$ylW. ü ü ü. $

xq$gq

$yqW

D @ â4$x4

$y4 . â5

$x5

$y5 . ü ü ü. âl$xl$yl . ü ü ü. $

xq$gq

$yqW

Zh nqrz wkdw wkh whupvqâ4

$x4

$y4> â5

$x5

$y5> ===> âq

$xq

$yq

rduh lq rughu ri grplqdqfh

iurp juhdwhvw wr ohdvw1 Wkhuhiruh/ dq dssur{lpdwlrq ri wkh pdwul{ D fdq ehdfklhyhg e| uhgxflqj wkh qxpehu ri lwhudwlrqv lqyroyhg lq wkh olqhdu frpelqdwlrq1

Dl @ â4$x4

$y4 . â5

$x5

$y5 . ü ü ü. âl$xl$yl

ô }ù?sùî @?_ |üi 5V#

Lw*v txlwh reylrxv wkdw wkh pdwkhpdwlfv ehklqg wkh vlqjxodu ydoxh ghfrpsrvlwlrqzrxog ehfrph h{wudruglqdulo| lqyroyhg udwkhu txlfno|1 Iru wklv uhdvrq/ rqfhwkh pdwkhpdwlfv kdyh ehhq xqghuvwrrg/ lw lv d jrrg lghd wr xvh d pdwkhpdwlfvvriwzduh1 Pdwode lv rqh h{dpsoh wkdw zrunv txlwh qlfho|1 Wr ñqg rxw pruh derxwwkhvh frppdqgv dqg rwkhuv zkloh zrunlqj lq Pdwode xvh wkh khos frppdqg1Iru h{dpsoh/ li wkh frppdqg lv olqvsdfh+3/8,/ w|sh khos olqvsdfh wr ñqgrxw pruh derxw wkh olqvsdfh frppdqg1

H{huflvh 8 Fuhdwh d udqgrp ; û 43 pdwul{ D zlwk d udqn ri 9 dqg lqwhjhuydoxhv udqjlqj iurp ý97 wr 971 Xvlqj }>z'>K*v pñ[ frppdqg/ ñqg wkh pdwulfhvX/ G/ ) Y fruuhvsrqglqj wr D1

Wr fuhdwh d pdwul{ ri udqgrp lqwhjhuv/ wkh hdvlhvw zd| lv wr xvh wkh udqglqwfrppdqg +|rx pxvw kdyh DWODVW iru wklv wr zrun,1 Wkh frppdqg zlwk wkhvhsdudphwhuv uhdgv=

AA D@udqglqw+;/43/97/9,

Wkhq/ doo wkdw lv uhtxluhg lv wkh delolw| wr w|sh wkh qh{w olqh lqwr wkh frp0sxwhu=

AA ^X/G/Y`@vyg+D,

Li |rx zlvk/ lw lv srvvleoh wr frqñup wkh udqn ri wkh pdwul{ D e| w|slqj wkhfrppdqg=

AA udqn+D,

Ru e| w|slqj

9

Page 7: The Singular Value Decomposition and ItYs Applications in Image Processing

AA gldj+G,

dqg vhhlqj wkdw wzr ri wkh vlqjxodu ydoxhv ri D duh }hur/ ru vr qhduo| }hur wkdwwkh| duh qhjoljleoh1 Dqg/ wkdw*v lw1 Wkh pdwul{ D zdv lqsxwwhg/ dqg wkhpdwulfhv X / G/ ) Y zhuh frpsxwhg1

e W4@}i èhLUittð?} @?_ |üi 5V#

Lq rughu wr xqghuvwdqg wkh iroorzlqj vhfwlrq/ d eulhi glvfxvvlrq derxw krzPdwode frqvwuxfwv lpdjhv iurp pdwulfhv lv qhfhvvdu|1 Edvlfdoo|/ hdfk hqwu|lq wkh pdwul{ fruuhvsrqgv wr d vpdoo vtxduh ri wkh lpdjh1 Wkh qxphulfdo ydoxhri wkh hqwu| fruuhvsrqgv wr d froru lq Pdwode1 Wkh froru vshfwuxp fdq ehvhhq lq Pdwode e| w|slqj wkh iroorzlqj frppdqgv/ dqg wkh slfwxuhg vfuhhq zloodsshdu +4,1

AA F@4=97>

AA lpdjh+F,

Zlwk wklv lq plqg/ hqwhulqj d 6 û 6 pdwul{ ri udqgrp lqwhjhuv vkrxog jlyh dslfwxuh ri qlqh vtxduh eorfnv frpsulvlqj rqh odujh eorfn1 Ixuwkhupruh/ wkh froruri hdfk lqglylgxdo eorfn zloo fruuhvsrqg wr wkh froru slfwxuhg dw wkdw qxphulfdoydoxh lq ñjxuh +4,1

AA V@udqglqw+6/6,

AA lpdjh+V,

:

Page 8: The Singular Value Decomposition and ItYs Applications in Image Processing

Lw kdv ehhq vkrzq wkdw dq| pdwul{ V fdq eh dssur{lpdwhg xvlqj wkh dohvvhu qxpehu ri lwhudwlrqv zkhq fdofxodwlqj wkh olqhdu frpelqdwlrqv ghñqlqj V1Wklv fdq eh vkrzq xvlqj wkh Pdwode frppdqg +dydlodeoh wkurxjk DWODVW,vyglpdjh1 vyglpdjh lv dq lqwhudfwlyh surjudp zklfk vkrzv wkh ruljlqdo lpdjhsurgxfhg e| V dqg wkh lpdjh surgxfhg e| wkh dssur{lpdwhg D zlwk wkh vkrzqqxpehu ri lwhudwlrqv1 Wkh iroorzlqj ghprqvwudwhv +ñjxuhv zloo ydu| gxh wr wkhxvh ri udqgrp lqwhjhuv,1

AA ^X/G/Y`@vyg+V,

AA vyglpdjh+V/X/G/Y,

Rqh Lwhudwlrq Wzr Lwhudwlrqv Wkuhh Lwhudwlrqv

Wkh ruljlqdo lpdjh zdv qrw rewdlqhg xqwlo wkh wklug lwhudwlrq/ exw wklv fdqeh h{sodlqhg e| w|slqj

AA udqn+D,

Ri frxuvh/ uhdol}h wkdw lq wkh vfrsh ri d pruh ghwdlohg lpdjh/ vd| 845 û 845wkh glvfrorulqj ri d sl{ho khuh ru wkhuh pljkw qrw eh ri frqfhuq1 Xvh Pdwodewr frqvwuxfw d 9 û 9 pdwul{ ri udqn 71 Wklv wlph doorz froru ydoxhv iurpehwzhhq ý97 dqg 971 Wkh ruljlqdo lpdjh vkrxog eh dfklhyhg e| wkh irxuwkdssur{lpdwlrq1 Phdqlqj wkdw wkh 7 û 7 pdwul{ Vu ri rqo| vl{whhq hqwulhv jlyhvwkh vdph lpdjh dv wkh 9 û 9 pdwul{ ri 69 hqwulhv1 D uhgxfwlrq ri pruh wkdq83( lq vl}h1

;

Page 9: The Singular Value Decomposition and ItYs Applications in Image Processing

AA V@udqglqw+9/9/97/7,

AA ^X/G/Y`@vyg+V,

AA vyglpdjh+V/X/G/Y,

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Page 10: The Singular Value Decomposition and ItYs Applications in Image Processing

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Page 11: The Singular Value Decomposition and ItYs Applications in Image Processing

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Page 12: The Singular Value Decomposition and ItYs Applications in Image Processing

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